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Game ideals. (English) Zbl 1173.03036

For \(\kappa\) a regular uncountable cardinal and \(\lambda\) a cardinal, \(\lambda>\kappa\), let \(A\subseteq P_{\kappa}(\lambda)\). The author considers the following two-player (I and II) game \(H_{\kappa, \lambda}(A)\): there are \(\omega\) moves and I goes first; I and II alternately choose members of \(P_{\kappa}(\lambda)\). The result is a sequence \(\langle a_n : n<\omega\rangle\). Player I wins if and only if \(\bigcup_{n<\omega} a_n\notin A\). The set of all \(A\) for which player I has a winning strategy in \(H_{\kappa, \lambda}(A)\) is a normal ideal on \(P_{\kappa}(\lambda)\), denoted by \(NG_{\kappa,\lambda}\). The paper under review studies the properties of \(NG_{\kappa,\lambda}\), using such tools as club-guessing, scales, Namba trees, and mutually stationary sets.
The author shows that \(P_{\kappa}(\lambda)\) is the disjoint union of \(M(\kappa,\lambda)\) sets not in \(NG_{\kappa,\lambda}\), where \(M(\kappa,\lambda)\) is \(\lambda^{\aleph_0}\) if \(\text{cf}(\lambda)=\omega\) or \(\text{cf}(\lambda)\geq\kappa\), and \(\lambda^{+}\cdot \lambda^{\aleph_0}\) otherwise. He also observes that if there is no inner model with \(\aleph_2\) measurable cardinals, then \(M(\kappa,\lambda)\) is the least size of any member of the dual filter \(NG_{\kappa,\lambda}^{*}\), and so \(P_{\kappa}(\lambda)\) cannot be decomposed into more than \(M(\kappa,\lambda)\) members of \(NG_{\kappa,\lambda}^{+}\). In the paper’s final section the author proves that if \(2^{<\kappa}\leq \mu^{\aleph_0}\) for some regular cardinal \(\mu\) with \(\kappa<\mu\leq\lambda\), then \(\diamondsuit_{\kappa,\lambda}[NG_{\kappa,\lambda}]\) holds.

MSC:

03E05 Other combinatorial set theory
03E02 Partition relations
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